Immersions of $C_2$-projective spaces via $K\mathbb{R}$-theory

TL;DR

This paper computes the Atiyah Real K-theory of $C_2$-equivariant projective spaces and constructs their immersions into regular representations, revealing an equivariant James periodicity.

Contribution

It introduces new computations of Atiyah Real K-theory for $C_2$-projective spaces and constructs explicit immersions, utilizing recent geometric filtrations and spectral sequences.

Findings

Computed Atiyah Real K-theory of $C_2$-projective spaces.

Constructed immersions into multiples of the regular representation.

Derived an equivariant analogue of James periodicity.

Abstract

We compute the Atiyah Real $K$-theory of $C_2$-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces due to Bhattacharya-Waugh-Zeng-Zou, together with a variant of the localized slice spectral sequence introduced by Meier-Shi-Zeng. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.